Approximation by polynomials and Blaschke products having all zeros on a circle

Abstract

We show that a nonvanishing analytic function on a sub-disc of the unit disc can be approximated by (a scalar multiple of) a Blaschke product whose zeros lie on a prescribed circle enclosing the sub-disc. We also give a new proof of the analogous classical result for polynomials. A connection is made to universality results for the Riemann zeta function.